A semi-infinite elastic string is initially at rest on the $x$-axis with $0 \leqslant x<\infty$. The transverse displacement of the string, $y(x, t)$, is governed by the partial differential equation

$$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$$

where $c$ is a positive real constant. For $t \geqslant 0$ the string is subject to the boundary conditions $y(0, t)=f(t)$ and $y(x, t) \rightarrow 0$ as $x \rightarrow \infty$.

(i) Show that the Laplace transform of $y(x, t)$ takes the form

$$\hat{y}(x, p)=\hat{f}(p) e^{-p x / c}$$

(ii) For $f(t)=\sin \omega t$, with $\omega \in \mathbb{R}^{+}$, find $\hat{f}(p)$ and hence write $\hat{y}(x, p)$ in terms of $\omega, c, p$ and $x$. Obtain $y(x, t)$ by performing the inverse Laplace transform using contour integration. Provide a physical interpretation of the result.